题名 | An augmented Lagrangian method for a class of inverse quadratic programming problems |
作者 | |
发表日期 | 2010 |
发表期刊 | Applied Mathematics and Optimization
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ISSN/eISSN | 0095-4616 |
卷号 | 61期号:1页码:57-83 |
摘要 | We consider an inverse quadratic programming (QP) problem in which the parameters in the objective function of a given QP problem are adjusted as little as possible so that a known feasible solution becomes the optimal one. We formulate this problem as a minimization problem with a positive semidefinite cone constraint and its dual is a linearly constrained semismoothly differentiable (SC ) convex programming problem with fewer variables than the original one. We demonstrate the global convergence of the augmented Lagrangian method for the dual problem and prove that the convergence rate of primal iterates, generated by the augmented Lagrange method, is proportional to 1/r, and the rate of multiplier iterates is proportional to 1/\sqrt{r}, where r is the penalty parameter in the augmented Lagrangian. As the objective function of the dual problem is a SC function involving the projection operator onto the cone of symmetrically semi-definite matrices, the analysis requires extensive tools such as the singular value decomposition of matrices, an implicit function theorem for semismooth functions, and properties of the projection operator in the symmetric-matrix space. Furthermore, the semismooth Newton method with Armijo line search is applied to solve the subproblems in the augmented Lagrange approach, which is proven to have global convergence and local quadratic rate. Finally numerical results, implemented by the augmented Lagrangian method, are reported. © 2009 Springer Science+Business Media, LLC. |
关键词 | Inverse optimization Newton method Quadratic programming Rate of convergence The augmented Lagrangian method The cone of positive semidefinite matrices |
DOI | 10.1007/s00245-009-9075-z |
URL | 查看来源 |
语种 | 英语English |
Scopus入藏号 | 2-s2.0-73349086204 |
引用统计 | |
文献类型 | 期刊论文 |
条目标识符 | https://repository.uic.edu.cn/handle/39GCC9TT/6634 |
专题 | 北师香港浸会大学 |
通讯作者 | Zhang,Liwei |
作者单位 | 1.United International College,Beijing Normal University,Hong Kong Baptist University,Zhuhai,China 2.School of Mathematical Sciences,Dalian University of Technology,Dalian 116024,China |
第一作者单位 | 北师香港浸会大学 |
推荐引用方式 GB/T 7714 | Zhang,Jianzhong,Zhang,Liwei. An augmented Lagrangian method for a class of inverse quadratic programming problems[J]. Applied Mathematics and Optimization, 2010, 61(1): 57-83. |
APA | Zhang,Jianzhong, & Zhang,Liwei. (2010). An augmented Lagrangian method for a class of inverse quadratic programming problems. Applied Mathematics and Optimization, 61(1), 57-83. |
MLA | Zhang,Jianzhong,et al."An augmented Lagrangian method for a class of inverse quadratic programming problems". Applied Mathematics and Optimization 61.1(2010): 57-83. |
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